Quadratic analysis

To avoid the dependency of mixing quality on quadrat size and the number of particles, Aubin et al. [31] proposed an alternative method based on the analysis of spatial point patterns. This method differs from quadrat analysis mentioned above in that it is based on the distance Xi from each point to the nearest event (tracer particle) for a chosen lattice of m sample points in the domain and therefore it does not require the studied region to be divided into quadrats. The variance of the point-event distances around distance % [Equation (5.14)] provides a means to evaluate the mixedness of a system with respect to a well-mixed criterion. This predefined criterion is set hy the distance Xr, which corresponds to the scale of segregation whereby two events are considered spatially close enough to be mixed. Inthis case the variance is defined via... 

Petersen C etal. (1998) Linear-quadratic analysis of tumor response to fractionated radiotherapy a study on human squamous cell carcinoma xenografts. Int J Radiat Biol 73(2) 197-205... 

Abstract. A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the Ca atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Angstrom. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic progrmnming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has local minima within 1.3-4.7A of the PDB geometry, with one exception that has an error of S.SA. 

Keywords, protein folding, tertiary structure, potential energy surface, global optimization, empirical potential, residue potential, surface potential, parameter estimation, density estimation, cluster analysis, quadratic programming... 

Harmonic analysis is an alternative approach to MD. The basic assumption is that the potential energy can be approximated by a sum of quadratic terms in displacements. 

Two quadratic equations in two variables can in general be solved only by numerical methods (see Numerical Analysis and Approximate Methods ). If one equation is of the first degree, the other of the second degree, a solution may be obtained by solving the first for one unknown. This result is substituted in the second equation and the resulting quadratic equation solved. 

When a reaction has many participants, which may be the case even of apparently simple processes like pyrolysis of ethane or synthesis of methanol, a factorial or other experimental design can be made and the data subjected to a re.spon.se. suiface analysis (Davies, Design and Analysis of Industrial Experiments, Oliver Boyd, 1954). A quadratic of this type for the variables X, Xo, and X3 is... 

The resulting data of the Box-Behnken design were used to formulate a statistically significant empirical model capable of relating the extent of sugar 3deld to the four factors. A commonly used empirical model for response surface analysis is a quadratic polynomial of the type... 

We must stress, however, that the Black-Halperin analysis has been conducted for only a single substance, namely, amorphous silica, and systematic studies on other materials should be done. The discovered numerical inconsistency may well turn out to be within the deviations of the heat capacity and conductivity from the strict linear and quadratic laws, repsectively. Finally, a controllable kinetic treatment of a time-dependent experiment would be necessary. 

We have carrried out an analysis of the multilevel structure of the tunneling centers that goes beyond a semiclassical picture of the formation of those centers at the glass transition, which was primarily employed in this chapter. These effects exhibit themselves in a deviation of the heat capacity and conductivity from the nearly linear and quadratic laws, respectively, that are predicted by the semiclassical theory. 

This classification problem can then be solved better by developing more suitable boundaries. For instance, using so-called quadratic discriminant analysis (QDA) (Section 33.2.3) or density methods (Section 33.2.5) leads to the boundaries of Fig. 33.2 and Fig. 33.3, respectively [3,4]. Other procedures that develop irregular boundaries are the nearest neighbour methods (Section 33.2.4) and neural nets (Section 33.2.9). 

Equation (33.10) is applied in what is called quadratic discriminant analysis (QDA). The equations can be shown to describe a quadratic boundary separating the regions where is minimal for the classes considered. 

W. Wu, Y. Mallet, B. Walczak, W. Penninckx, D.L. Massart, S. Heuerding and F. Erni, Comparison of regularized discriminant analysis, linear discriminant analysis and quadratic discriminant analysis, applied to NIR data. Anal. Chim. Acta, 329 (1996) 257-265. 

The selection of the number of PLS-components to be included in the model was done according to the PRESS criterion (Section 36.3). Note that the result is comparable to the one which we obtained earlier by means of the simple Hansch analysis (Section 37.1.1). Hence, in this case, there is no obvious benefit to include a quadratic term of log P in the model. 

The experimental designs discussed in Chapters 24-26 for optimization can be used also for finding the product composition or processing condition that is optimal in terms of sensory properties. In particular, central composite designs and mixture designs are much used. The analysis of the sensory response is usually in the form of a fully quadratic function of the experimental factors. The sensory response itself may be the mean score of a panel of trained panellists. One may consider such a trained panel as a sensitive instrument to measure the perceived intensity useful in describing the sensory characteristics of a food product. 

By automation one can remove the variation of the analysis time or shorten the analysis time. Although the variation of the analysis time causes half of the delay, a reduction of the analysis time is more important. This is also true if, by reducing the analysis time, the utilization factor would remain the same (and thus q) because more samples are submitted. Since p = AT / lAT, any measure to shorten the analysis time will have a quadratic effect on the absolute delay (because vv = AT / (LAT - AT)). As a consequence the benefit of duplicate analyses (detection of gross errors) and frequent recalibration should be balanced against the negative effect on the delay. 

From the above analysis, it is found that both and IFqi are linear functions of the surface electric field strength E, so that A W obtained from Eq. (9) is also a function of E. The important point is that the z-dependence of the matrix element can be expressed only through E. By actually solving the secular equation (9), it has been found that the E-dependence of A IF is well represented by a quadratic equation ... 

The last two results are rather similar to the quadratic forms given by Fox and Uhlenbeck for the transition probability for a stationary Gaussian-Markov process, their Eqs. (20) and (22) [82]. Although they did not identify the parity relationships of the matrices or obtain their time dependence explicitly, the Langevin equation that emerges from their analysis and the Doob formula, their Eq. (25), is essentially equivalent to the most likely terminal position in the intermediate regime obtained next. 

The first energy moment of the isolated system is not conserved and it fluctuates about zero. According to the general analysis of Section IIB, the entropy of the isolated system may be written as a quadratic form,... 

The Grashof number given by Eq. (40) appears to have a weaker theoretical basis than that given by Eq. (37), since it is based on an analysis that approximates the profile of the vertical velocity component in free convection, for example, by a quadratic function of the distance to the electrode. The choice of an appropriate Grashof number, as well as the experimental conditions in the work of de Leeuw den Bouter et al. (DIO) and Marchiano and Arvia (M3), has been reviewed critically by Wragg and Nasiruddin (W10). They measured mass transfer by combined thermal and diffusional, turbulent, free convection at a horizontal plate [see Eq. (31) in Table VII], and correlated their results satisfactorily with the Grashof number of Eq. (37). 

Draper and Smith [1] discuss the application of DW to the analysis of residuals from a calibration their discussion is based on the fundamental work of Durbin, et al in the references listed at the beginning of this chapter. While we cannot reproduce their entire discussion here, at the heart of it is the fact that there are many kinds of serial correlation, including linear, quadratic and higher order. As Draper and Smith show (on p. 64), the linear correlation between the residuals from the calibration data and the predicted values from that calibration model is zero. Therefore if the sample data is ordered according to the analyte values predicted from the calibration model, a statistically significant value of the Durbin-Watson statistic for the residuals in indicative of high-order serial correlation, that is nonlinearity. 

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